Algebra 2 (1st Edition)

$2023203$
We know that if we want to select $r$ objects out of $n$ disregarding the order, we can do this in $_nC_r=\frac{n!}{r!(n-r)!}$ ways. Here we can count the number of possibilities by subtracting the number of options when we have $0$ spades from the number of possibilities. Hence here we have $n_1=52,r_1=5$ for all the possibilities and $n_2=39,r_2=5$ for when we have no spades. Hence the answer: $_{52}C_5-{39}C_5=\frac{52!}{47!5!}-\frac{39!}{34!5!}=2598960-575757=2023203$